Question

How do you order the following from least to greatest `4 4/5, sqrt19, -sqrt5, -21, sqrt5`?

Answer 1

`-21<-sqrt(5)< sqrt(5)<4 4/5 < sqrt(19)`

Explanation:

Notice that `-21` and `-sqrt(5)` are negative

or

`-21<0` and `-sqrt(5)<0`

One way to think of this when it comes to the order of these two negatives is that

`-21<-20` therefore `-sqrt(21)<-sqrt(20)=-(sqrt(4(5)))=-(sqrt(4)(sqrt(5)))=-2sqrt(5)`

So

`-21<-2sqrt(5)< -sqrt(5)`

Then

`-21< -sqrt(5)`

That takes care of the negative numbers, but we still have the three positive numbers.

First notice

`4 4/5= 16/5=3.5`

and also

`4<5<9`

`<=>`

`sqrt(4)< sqrt(5)< sqrt(9)` Take the square root of all sides

`<=>`

`2< sqrt(5) < 3`

Therefore `sqrt(5)< 4 4/5`

`4 4/5=3.5 => (4 4/5)^2=(3.5)^2`

`=(3+0.5)^2=9+2(3)(0.5)+0.25=9+3+0.25=12.25`

So we know that

`4 4/5=sqrt(12.25)`

and since

`12.25<19`

then

`sqrt(12.25) < sqrt(19)`

So we get

`ul(-21 < -sqrt(5) < sqrt(5) <4 4/5 < sqrt(19))`